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An *Asian* option is a path-dependent option
with a payoff linked to the average value of the underlying asset
during the life (or some part of the life) of the option. They are
similar to lookback options in that there are two types of Asian options:
fixed (average price option) and floating (average strike option).
Fixed Asian options have a specified strike, while floating Asian
options have a strike equal to the average value of the underlying
asset over the life of the option.

There are four Asian option types, each with its own characteristic payoff formula:

Fixed call (average price option): $$\mathrm{max}(0,{S}_{av}-X)$$

Fixed put (average price option): $$\mathrm{max}(0,X-{S}_{av})$$

Floating call (average strike option): $$\mathrm{max}(0,S-{S}_{av})$$

Floating put (average strike option): $$\mathrm{max}(0,{S}_{av}-S)$$

where:

is the average price of underlying asset.

is the price of the underlying asset.

is the strike price (applicable only to fixed Asian options).

is defined using either a geometric or an arithmetic average.

The following functions support Asian options.

Function | Purpose |
---|---|

Price European or American Asian options using the Longstaff-Schwartz model. | |

Calculate prices and sensitivities of European or American Asian options using the Longstaff-Schwartz model. | |

Price European geometric Asian options using the Kemna Vorst model. | |

Calculate prices and sensitivities of European geometric Asian options using the Kemna Vorst model. | |

Price European arithmetic Asian options using the Levy model. | |

Calculate prices and sensitivities of European arithmetic Asian options using the Levy model. |

A *vanilla option* is a category of options
that includes only the most standard components. A vanilla option
has an expiration date and straightforward strike price. American-style
options and European-style options are both categorized as vanilla
options.

The payoff for a vanilla option is as follows:

For a call: $$\mathrm{max}(St-K,0)$$

For a put: $$\mathrm{max}(K-St,0)$$

where:

*St* is the price of the underlying asset at
time *t*.

*K* is the strike price.

The following functions support specifying or pricing a vanilla option.

Function | Purpose |
---|---|

Price European, Bermudan, or American vanilla options using the Longstaff-Schwartz model. | |

Calculate European, Bermudan, or American vanilla option prices and sensitivities using the Longstaff-Schwartz model. | |

Calculate vanilla option prices using finite difference method. | |

Calculate vanilla option prices and sensitivities using finite difference method. |

A *spread option* is an option written
on the difference of two underlying assets. For example, a European
call on the difference of two assets *X1* and *X2* would
have the following pay off at maturity:

$$\mathrm{max}(X1-X2-K,0)$$

where:

*K* is the strike price.

The following functions support spread options.

Function | Purpose |
---|---|

Price European spread options using the Kirk pricing model. | |

Calculate European spread option prices and sensitivities using the Kirk pricing model. | |

Price European spread options using the Bjerksund-Stensland pricing model. | |

Calculate European spread option prices and sensitivities using the Bjerksund-Stensland pricing model. | |

Price European or American spread options using the Alternate Direction Implicit (ADI) finite difference method. | |

Calculate price and sensitivities of European or American spread options using the Alternate Direction Implicit (ADI) finite difference method. | |

Price European or American spread options using Monte Carlo simulations. | |

Calculate price and sensitivities for European or American spread options using Monte Carlo simulations. |

For more information on using spread options, see Pricing European and American Spread Options.

A *lookback* option is
a path-dependent option based on the maximum or minimum value the
underlying asset (e.g. electricity, stock) achieves during the entire
life of the option. Basically the holder of the option can ‘look
back’ over time to determine the payoff. This type of option
provides price protection over a selected period, reduces uncertainties
with the timing of market entry, moderates the need for the ongoing
management, and therefore, is usually more expensive than vanilla
options.

Lookback call options give the holder the right to buy the underlying asset at the lowest price. Lookback put options give the right to sell the underlying asset at the highest price.

Financial Instruments Toolbox™ software supports two types of lookback options: fixed and floating. The difference is related to how the strike price is set in the contract. Fixed lookback options have a specified strike price and the option pays out the maximum of the difference between the highest (lowest) observed price of the underlying during the life of the option and the strike. Floating lookback options have a strike price determined at maturity, and it is set at the lowest (highest) price of the underlying reached during the life of the option. This means that for a floating strike lookback call (put), the holder has the right to buy (sell) the underlying asset at its lowest (highest) price observed during the life of the option. So, there are a total of four lookback option types, each with its own characteristic payoff formula:

Fixed call: $$\mathrm{max}(0,{S}_{\mathrm{max}}-X)$$

Fixed put: $$\mathrm{max}(0,X-{S}_{\mathrm{min}})$$

Floating call: $$\mathrm{max}(0,S-{S}_{\mathrm{min}})$$

Floating put: $$\mathrm{max}(0,{S}_{\mathrm{max}}-S)$$

where:

$${S}_{\mathrm{max}}$$ is the maximum price of underlying asset.

is the minimum price of underlying asset.

is the price of the underlying asset at maturity.

is the strike price.

The following functions support lookback options.

Function | Purpose |
---|---|

Calculate prices of European lookback fixed and floating strike options using the Conze-Viswanathan and Goldman-Sosin-Gatto models. | |

Calculate prices and sensitivities of European fixed and floating strike lookback options using the Conze-Viswanathan and Goldman-Sosin-Gatto models. | |

Calculate prices of lookback fixed and floating strike options using the Longstaff-Schwartz model. | |

Calculate prices and sensitivities of lookback fixed and floating strike options using the Longstaff-Schwartz model. |

Lookback options and Asian options are instruments used in the electricity market to manage purchase timing risk. Electricity purchasers cover part of their expected electricity consumption on the forward market to avoid the volatility and limited liquidity of the spot market. Using Asian options as a hedging tool is a passive approach to solving the purchase timing problem. An Asian option instrument diminishes the wrong timing risk but it also reduces any potential benefit to the buyer from falling prices. On the other hand, lookback options allow the purchasers to buy electricity at the lowest price, but as mentioned before, this instrument is more expensive than Asian and vanilla options.

A *forward option* is a non-standardized
contract between two parties to buy or to sell an asset at a specified
future time at a price agreed upon today. The buyer of a forward option
contract has the right to hold a particular forward position at a
specific price any time before the option expires. The forward option
seller holds the opposite forward position when the buyer exercises
the option. A call option is the right to enter into a long forward
position and a put option is the right to enter into a short forward
position. A closely related contract is a futures contract. A forward
is like a futures in that it specifies the exchange of goods for a
specified price at a specified future date. The table below displays
some of the characteristics of forward and futures contracts.

Forwards | Futures |
---|---|

Customized contracts | Standardized contracts |

Over the counter traded | Exchange traded |

Exposed to default risk | Clearing house reduces default risk |

Mostly used for hedging | Mostly used by hedgers and speculators |

Settlement at the end of contract (no Margin required) | Daily changes are settled day by day (Margin required) |

Delivery usually takes place | Delivery usually never happens |

The payoff for a forward option, where the value of a forward
position at maturity depends on the relationship between the delivery
price (*K*) and the underlying price (*S** _{T}*)
at that time, is:

For a long position: $${f}_{T}={S}_{T}-K$$

For a short position: $${f}_{T}=K-{S}_{T}$$

The following functions support pricing a forwards option.

Function | Purpose |
---|---|

Price options on forwards using the Black option pricing model. | |

Determine option prices and sensitivities on forwards using the Black pricing model. |

A *future option* is a standardized contract
between two parties to buy or sell a specified asset of standardized
quantity and quality for a price agreed upon today (the futures price)
with delivery and payment occurring at a specified future date, the
delivery date. The contracts are negotiated at a futures exchange,
which acts as an intermediary between the two parties. The party agreeing
to buy the underlying asset in the future, the "buyer" of the contract,
is said to be "long", and the party agreeing to sell the asset in
the future, the "seller" of the contract, is said to be "short."

Forwards | Futures |
---|---|

Customized contracts | Standardized contracts |

Over the counter traded | Exchange traded |

Exposed to default risk | Clearing house reduces default risk |

Mostly used for hedging | Mostly used by hedgers and speculators |

Settlement at the end of contract (no Margin required) | Daily changes are settled day by day (Margin required) |

Delivery usually takes place | Delivery usually never happens |

A futures contract is the delivery of item *J* at
time *T* and:

There exists in the market a quoted price $$F(t,T)$$, which is known as the futures price at time

*t*for delivery of*J*at time*T*.The price of entering a futures contract is equal to zero.

During any time interval [

*t*,*s*], the holder receives the amount $$F(s,T)-F(t,T)$$ (this reflects instantaneous marking to market).At time

*T*, the holder pays $$F(T,T)$$ and is entitled to receive*J*. Note that $$F(T,T)$$ should be the spot price of*J*at time*T*.

The following functions support pricing a futures option.

Function | Purpose |
---|---|

Price options on futures using the Black option pricing model. | |

Determine option prices and sensitivities on futures using the Black pricing model. |

`asianbykv`

| `asianbylevy`

| `asianbyls`

| `asiansensbykv`

| `asiansensbylevy`

| `asiansensbyls`

| `lookbackbycvgsg`

| `lookbackbyls`

| `lookbacksensbycvgsg`

| `lookbacksensbyls`

| `optpricebysim`

| `optstockbyblk`

| `optstockbyls`

| `optstocksensbyblk`

| `optstocksensbyls`

| `spreadbybjs`

| `spreadbyfd`

| `spreadbykirk`

| `spreadbyls`

| `spreadsensbybjs`

| `spreadsensbyfd`

| `spreadsensbykirk`

| `spreadsensbyls`

- Pricing European and American Spread Options
- Hedging Strategies Using Spread Options
- Pricing Swing Options using the Longstaff-Schwartz Method
- Compute Option Prices on a Forward
- Compute Forward Option Prices and Delta Sensitivities
- Compute the Option Price on a Future
- Simulating Electricity Prices with Mean-Reversion and Jump-Diffusion
- Pricing Asian Options

- Forwards Option
- Futures Option
- Spread Option
- Asian Option
- Vanilla Option
- Lookback Option
- Supported Equity Derivatives
- Supported Interest-Rate Instruments

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